# Review Notes for Linear Algebra True or False Last Updated: February 22, 2010

Size: px
Start display at page:

Download "Review Notes for Linear Algebra True or False Last Updated: February 22, 2010"

## Transcription

1 Review Notes for Linear Algebra True or False Last Updated: February 22, 2010 Chapter 4 [ Vector Spaces 4.1 If {v 1,v 2,,v n } and {w 1,w 2,,w n } are linearly independent, then {v 1 +w 1,v 2 +w 2,,v n +w n } is linearly Let {e 1,e 2 } be the standard basis of R 2. Take {v 1,v 2 } = {e 1,e 2 } and {w 1,w 2 } = { e 1, e 2 }. Then {v 1,v 2 } and {w 1,w 2 } are linearly But then {v 1 + w 1,v 2 + w 2 } = {0,0} is of course linearly dependent. 4.2 If {v 1,v 2,,v n } and {w 1,w 2,,w n } are linearly dependent, then {v 1 +w 1,v 2 +w 2,,v n +w n } is linearly dependent. Take {v 1,v 2 } = {e 1,0} and {w 1,w 2 } = {0,e 2 }. Then {v 1,v 2 } is linearly dependent because 0 e = 0, and {w 1,w 2 } is linearly dependent because e 2 = 0. But then {v 1 + w 1,v 2 + w 2 } = {e 1,e 2 } is of course linearly 4.3 If {v 1,v 2,,v n } is linearly independent, and {v n,v n+1,,v n+k } is linearly independent, then {v 1,v 2,,v n+k } is linearly Take {v 1,v 2 } = {(1, 0),(0, 1)} and {v 2,v 3 } = {(0, 1), (1,1)}. Then v 3 = v 1 + v If {v 1,v 2,,v n+k } is linearly independent, then {v 1,v 2,,v n } is linearly True. If any nonempty subset of S = {v 1,v 2,,v n+k } is linearly dependent, then S must also be linearly dependent. 4.5 If {v 1,v 2,,v n+k } is linearly dependent, then {v 1,v 2,,v n } is linearly dependent. {(0, 1), (1, 0), (1, 1)} are linearly dependent but {(0, 1), (1, 0)} are linearly 4.6 If any two of v 1, v 2, v 3 are linearly independent, then the three vectors v 1, v 2, v 3 are linearly Take v 1 = (1,0), v 2 = (0,1), v 3 = (1, 1). Then three vectors in R 2 must be linearly dependent. 4.7 If the three vectors v 1, v 2, v 3 are linearly independent, then any two of v 1, v 2, v 3 are linearly True. Follows from Let S be a set of k vectors in R n. If k < n, then S is linearly Take v 1 = (1,1, 1), v 2 = (2,2, 2) in R 3. Then S = {v 1,v 2 } is linearly dependent. 4.9 If rank A m n = m, then the columns of A are linearly [ 1 A = with ranka = 2. A has a zero column = not all columns of A are pivot = columns of A are linearly dependent If rank A m n = n, then the columns of A are linearly True. rank A = n = all columns of A are pivot = columns of A are linearly 1

2 4.11 If rank A m n = m, then the rows of A are linearly True. rank A = m = all rows of A are pivot = rows of A are linearly 4.12 If rank A m n = n, then the rows of A are linearly A = with ranka = 2. A has a zero row = not all rows of A are pivot = rows of A are linearly dependent The columns of a matrix A are linearly independent if the equation Ax = 0 has the trivial solution. if and only if the equation Ax = 0 has only the trivial solution. The columns of a matrix A are linearly independent 4.14 The columns of any 4 5 matrix are linearly If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v 1,v 2,,v m} in R n is linearly dependent if m > n If rank A m n = m, then the columns of A span R m. True. rank A = m = all rows of A are pivot = Ax = b has solutions for all b R m = b is a linear combination of columns of A for all b = columns of A span R m If rank A m n = n, then the columns of A span R m. A = with ranka = 2. We can find b = R 3 such that 1 b is not a linear combination of 1 0, 0 1 = columns of A do not span R If b is in the span of {v 1,v 2,,v n+1 }, then b is in the span of {v 1,v 2,,v n }. Let {e 1,e 2 } be the standard basis of R 2. Then (1,1) span {e 1,e 2 } but (1,1) span {e 1 } If b is in the span of {v 1,v 2,,v n }, then b is in the span of {v 1,v 2,,v n+1 }. True. b = a 1 v a nv n for some a i. Then b = a 1 v a nv n + 0v n If v 1, v 2,, v n are in R m, then span {v 1,v 2,,v n } is the same as the column space of the matrix [ v1 v 2 v n. True. If A = [ v 1 v 2 v n, with the columns in R m, then ColA is the same as span {v 1,v 2,,v n}. The column space of an m n matrix is a subspace of R m The row space of A is the same as the column space of A t. True. If A is an m n matrix, each row of A has n entries and thus can be identified with a vector in R n. The set of all linear combinations of the row vectors is called the row space of A and is denoted by RowA. Each row has n entries, so RowA is a subspace of R n. Since the rows of A can be identified with the columns of A t, we could also write ColA t in place of RowA. 2

3 4.21 If v 1, v 2,, v n span R n, then {v 1,v 2,,v n } is linearly True. Take A = [ v 1 v 2 v n. Then A is square. Now, v1, v 2,, v n span R n = all rows of A are pivot = ranka = n = all columns of A are pivot = {v 1,v 2,,v n} is linearly 4.22 If v 1, v 2,, v n span R n, then {v 1,v 2,,v n } is a basis of R n. True. Follows from If columns of A 5 3 are linearly independent, then rows of A span R 3. True. Columns of A are linearly independent = all columns of A are pivot = all rows of A t are pivot = ranka t = 3. Then for any b R 3, rank [ A t b 3 6 = rankat = 3 = b is a linear combination of columns of A t for any b R 3 = b is a linear combination of rows of A for any b R 3 = rows of A span R If v 1, v 2,, v n span V, then dim V n. True. If {v 1,v 2,,v n} is linearly independent, then {v 1,v 2,,v n} is a basis of the span V. Thus, dim V = number of vectors in a basis of V = n. If {v 1,v 2,,v n} is linearly dependent, then dim V < n If v 1, v 2,, v n span V, then v 1, v 2,, v n, v n+1,, v n+k span V. True. For b V, there exist scalars c 1, c 2,, c n such that b = c 1 v 1 + c 2 v c nv n = c 1 v 1 + c 2 v c nv n + 0v n v n+k = b can be expressed as a linear combination of v 1, v 2,, v n, v n+1,, v n+k. In other words, v 1, v 2,, v n, v n+1,, v n+k span V If v 1, v 2,, v n do not span V, then v 1, v 2,, v n, v n+1 do not span V. v 1 = (1, 0,0), v 2 = (0, 1,0) do not span R 3, but v 1 = (1,0, 0), v 2 = (0,1, 0), v 3 = (0,0, 1) span R If v 1, v 2,, v n span V, then {v 1,v 2,,v n } is a basis of V. The vectors can be linearly dependent. Let {e 1, e 2, e 3 } be the standard basis of R 3. Then {e 1, e 2, e 1 + e 2 } span a plane in R 3 but {e 1, e 2, e 1 + e 2 } is not a basis (vectors being linearly dependent) If {v 1,v 2,,v n } is a basis of R n and A n n is invertible, then {Av 1,Av 2,,Av n } is also a basis of R n. True. Let B n n = [ [ [ Av 1 Av n = A v1 v n = AT, where T = v1 v n. By given, n n {v 1,,v n} is a basis of R n = all rows as well as all columns of T are pivot = T is row equivalent to I n = T is invertible. Hence, B is also invertible. It follows that, for Av 1,,Av n R n, {Av 1,Av 2,,Av n} is a basis of R n The functions 1, sint, cos t are linearly True. Suppose c 1 + c 2 sin t + c 3 cos t = 0. Then t = 0 = c 1 + c 3 = 0; t = π/2 = c 1 + c 2 = 0; t = π = c 1 c 3 = 0. Hence, c 1 = c 2 = c 3 = 0. By definition, the functions are linearly 4.30 The functions 1, sin 2 t, cos 2 t are linearly Linear dependence relation is: 1 sin 2 t cos 2 t = 0. 3

4 4.31 A homogeneous equation is always consistent. True. Since A0 = 0, the equation Ax = 0 always has the zero solution x = If x is a nontrivial solution of Ax = 0, then every entry in x is nonzero. [ 1 x = 0 has a nontrivial solution x = The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of R m. The null space of an m n matrix A is a subspace of R n. Equivalently, the set of all solutions of a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace of R n The columns of an invertible n n matrix from a basis for R n. True. The columns of an invertible n n matrix from a basis for R n because they are linearly independent ( all columns of A are pivot) and span R n ( all rows of A are pivot) If ranka 7 5 = 5, then the dimension of the null space of A is 0. True. ranka = 5 = all columns of A are pivot = no nonpivot column of A = no free variable = Ax = 0 has only trivial solution = NulA def. = {x : Ax = 0} = {0} = NulA has no basis = dim (NulA) = The dimension of NulA is the number of variables in the equation Ax = b. By definition, the dimension of NulA is the number of vectors in a basis of NulA. The number of vectors in a basis of NulA is equal to the number of nonpivot columns of A which must be strictly smaller than the number of columns of A (in other words, which must be strictly smaller than the number of variables) The dimension of ColA is the number of pivot columns of A. the dimension of ColA is just the number of pivot columns of A, that is the rank of A. True. Since the pivot columns of A form a basis for ColA, 4.38 The dimensions of ColA and NulA add up to the number of columns of A. True. The nonpivot columns of A correspond to the free variables in Ax = 0. Thus the dimension of Nul A is the number of nonpivot columns of A. Since the number of pivot columns plus the number of nonpivot columns of A is exactly the number of columns, the dimensions of Col A and Nul A have the useful connection: If a matrix A has n columns, then rank A + dim Nul A = n If Ax = 0 has only trivial solution, then the columns of A form a basis of ColA. True. Ax = 0 has only trivial solution = no free variable = no nonpivot column of A = all columns of A are pivot = columns of A are linearly independent = columns of A form a basis of ColA If Ax = 0 has only trivial solution, then the rank of A is the number of columns of A. True. Ax = 0 has only trivial solution = all columns of A are pivot = ranka = number of columns of A. 4

5 4.41 If Ax = 0 has only trivial solution, then the rank of A is the number of rows of A. A = Then Ax = 0 has only trivial solution but ranka = If the columns of an m n matrix A span R m, then the equation Ax = b is consistent for each b in R m. True. Columns of A span R m = b is a linear combination of columns of A for all b R m = Ax = b has solutions for all b R m If A is an m n matrix whose columns do not span R m, then the equation Ax = b is inconsistent for some b in R m. True. Columns of A do not span R m = Ax = b has no solution for some b R m. = b cannot not be a linear combination of columns of A for some b R m 4.44 If A is an m n matrix and if the equation Ax = b is inconsistent for some b in R m, then A cannot have a pivot position in every row. = not all rows of A are pivot. True. Ax = b has no solution for some b in R m = b is pivot in [ A b for some b in R m 4.45 If Ax = b has solutions for all b, then the columns of A form a basis of ColA. [ [ b1 A = and b =. Then Ax = b has solutions for all b but columns of A are linearly dependent b 2 (column 3 = column 1 + column 2). Thus, columns of A do not form a basis of ColA If B is a row echelon form of a matrix A, then the pivot columns of B form a basis for ColA. A = = B. 1 0 However, the pivot columns (the first and third) of B do not span ColA because, for example, (0, 0,1) cannot be a linear combination of the pivot columns of B If Ax = b has solutions for all b, then the rank of A is the number of columns of A. Ax = b has solutions for all [ b = all rows of A are pivot = ranka = number of rows of A 1 ranka = number of columns of A. Take A = If Ax = b has solutions for all b, then the rank of A is the number of rows of A. True. Ax = b has solutions for all b = all rows of A are pivot = ranka = number of rows of A If [ A b is an invertible 4 4 matrix, then b is in the column space of A. [ [ A b is invertible = A b is row equivalent to I = b is pivot = Ax = b has no solution = b is not a linear combination of columns of A = b ColA If [ A b is an 3 5 matrix with ranka = 3, then b is in the column space of A. True. rank A = 3 = all rows of A are pivot = Ax = b has solutions for all b = b Col A. 5

6 4.51 If [ A b is an 4 4 matrix with rank [ A b = ranka, then b is in the column space of A. True. rank [ A b = ranka = b is nonpivot = Ax = b has solutions = b ColA If the vectors v 1, v 2, v 3, v 4 in R 4 are linearly dependent, then the 4 4 matrix A = [ v 1 v 2 v 3 v 4 is not invertible. True. Columns of A are linearly dependent = not all columns of A are pivot = ranka < 4 = row echelon form of A has a zero row = deta = 0 (by 2.31) = A is not invertible If the 4 4 matrix A = [ v 1 v 2 v 3 v 4 is not invertible, then the vectors v1, v 2, v 3, v 4 in R 4 are linearly dependent. True. A is not invertible = A is not row equivalent to I = row echelon form of A has a zero row = ranka < 4 = not all columns of A are pivot = columns of A are linearly dependent If A is an 3 3 matrix with ranka = 3, then ColA = R 3. b ColA for all b R 3 = ColA R 3 = ColA = R 3. True. ranka = 3 = all rows of A are pivot = columns of A span R 3 = 4.55 If A is an 3 3 matrix with ranka = 3, then RowA = R 3. True. ranka = 3 = ranka t = 3 = ColA t = R 3 (by 7.54) = RowA = R If v 1 and v 2 are in R 4 and v 1 is not a scalar multiple of v 2, then {v 1,v 2 } is linearly Choose v 1 0 and v 2 = If v 1, v 2, v 3, v 4 are vectors in R 4 and {v 1,v 2,v 3 } is linearly independent, then {v 1,v 2,v 3,v 4 } is linearly Choose v 4 = v If v 1, v 2, v 3, v 4 are linearly independent vectors in R 4, then {v 1,v 2,v 3 } is linearly True. Refer to If V has a basis of n vectors, then every basis of V must consist of exactly n vectors. True. Let B = {b 1,b 2,,b n} be a basis of V. Suppose that B 1 = {d 1,d 2,,d m} is another basis of V. Recall that a basis is a maximal independent set as well as a minimal spanning set. Now, since B 1 is linearly independent, we have m n. On the other hand, B 1 is a basis and B is linearly independent, we have n m as well. Thus, m = n The span of any vector in R 3 is a line. v = 0 = span {v} = {0} contains origin only The span of any two nonzero vectors in R 3 is a plane. Nonzero u, v in R 3 such that u v = span {u,v} = span {u} = a line. 6

### Chapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer.

Chapter 3 Directions: For questions 1-11 mark each statement True or False. Justify each answer. 1. (True False) Asking whether the linear system corresponding to an augmented matrix [ a 1 a 2 a 3 b ]

### 2018 Fall 2210Q Section 013 Midterm Exam II Solution

08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique

### Practice Final Exam. Solutions.

MATH Applied Linear Algebra December 6, 8 Practice Final Exam Solutions Find the standard matrix f the linear transfmation T : R R such that T, T, T Solution: Easy to see that the transfmation T can be

### 1. Determine by inspection which of the following sets of vectors is linearly independent. 3 3.

1. Determine by inspection which of the following sets of vectors is linearly independent. (a) (d) 1, 3 4, 1 { [ [,, 1 1] 3]} (b) 1, 4 5, (c) 3 6 (e) 1, 3, 4 4 3 1 4 Solution. The answer is (a): v 1 is

### Math 102, Winter 2009, Homework 7

Math 2, Winter 29, Homework 7 () Find the standard matrix of the linear transformation T : R 3 R 3 obtained by reflection through the plane x + z = followed by a rotation about the positive x-axes by 6

### (a) only (ii) and (iv) (b) only (ii) and (iii) (c) only (i) and (ii) (d) only (iv) (e) only (i) and (iii)

. Which of the following are Vector Spaces? (i) V = { polynomials of the form q(t) = t 3 + at 2 + bt + c : a b c are real numbers} (ii) V = {at { 2 + b : a b are real numbers} } a (iii) V = : a 0 b is

### Kevin James. MTHSC 3110 Section 4.3 Linear Independence in Vector Sp

MTHSC 3 Section 4.3 Linear Independence in Vector Spaces; Bases Definition Let V be a vector space and let { v. v 2,..., v p } V. If the only solution to the equation x v + x 2 v 2 + + x p v p = is the

### Solutions to Section 2.9 Homework Problems Problems 1 5, 7, 9, 10 15, (odd), and 38. S. F. Ellermeyer June 21, 2002

Solutions to Section 9 Homework Problems Problems 9 (odd) and 8 S F Ellermeyer June The pictured set contains the vector u but not the vector u so this set is not a subspace of The pictured set contains

### Study Guide for Linear Algebra Exam 2

Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real

### MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.

MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:

### Math 54 HW 4 solutions

Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,

### Announcements Monday, October 29

Announcements Monday, October 29 WeBWorK on determinents due on Wednesday at :59pm. The quiz on Friday covers 5., 5.2, 5.3. My office is Skiles 244 and Rabinoffice hours are: Mondays, 2 pm; Wednesdays,

### Math Final December 2006 C. Robinson

Math 285-1 Final December 2006 C. Robinson 2 5 8 5 1 2 0-1 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the

### Math 369 Exam #2 Practice Problem Solutions

Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.

### EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016

EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016 Answer the questions in the spaces provided on the question sheets. You must show your work to get credit for your answers. There will

### Math 2174: Practice Midterm 1

Math 74: Practice Midterm Show your work and explain your reasoning as appropriate. No calculators. One page of handwritten notes is allowed for the exam, as well as one blank page of scratch paper.. Consider

### 1 Systems of equations

Highlights from linear algebra David Milovich, Math 2 TA for sections -6 November, 28 Systems of equations A leading entry in a matrix is the first (leftmost) nonzero entry of a row. For example, the leading

### MATH 2210Q MIDTERM EXAM I PRACTICE PROBLEMS

MATH Q MIDTERM EXAM I PRACTICE PROBLEMS Date and place: Thursday, November, 8, in-class exam Section : : :5pm at MONT Section : 9: :5pm at MONT 5 Material: Sections,, 7 Lecture 9 8, Quiz, Worksheet 9 8,

### Vector Spaces 4.3 LINEARLY INDEPENDENT SETS; BASES Pearson Education, Inc.

4 Vector Spaces 4.3 LINEARLY INDEPENDENT SETS; BASES LINEAR INDEPENDENT SETS; BASES An indexed set of vectors {v 1,, v p } in V is said to be linearly independent if the vector equation c c c 1 1 2 2 p

### Math 3191 Applied Linear Algebra

Math 191 Applied Linear Algebra Lecture 16: Change of Basis Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/0 Rank The rank of A is the dimension of the column space

### The definition of a vector space (V, +, )

The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element

### ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST

me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following

### What is on this week. 1 Vector spaces (continued) 1.1 Null space and Column Space of a matrix

Professor Joana Amorim, jamorim@bu.edu What is on this week Vector spaces (continued). Null space and Column Space of a matrix............................. Null Space...........................................2

### Math 54. Selected Solutions for Week 5

Math 54. Selected Solutions for Week 5 Section 4. (Page 94) 8. Consider the following two systems of equations: 5x + x 3x 3 = 5x + x 3x 3 = 9x + x + 5x 3 = 4x + x 6x 3 = 9 9x + x + 5x 3 = 5 4x + x 6x 3

### Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017

Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...

### DEPARTMENT OF MATHEMATICS

DEPARTMENT OF MATHEMATICS. Points: 4+7+4 Ma 322 Solved First Exam February 7, 207 With supplements You are given an augmented matrix of a linear system of equations. Here t is a parameter: 0 4 4 t 0 3

### We showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true.

Dimension We showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true. Lemma If a vector space V has a basis B containing n vectors, then any set containing more

### YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions

YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For

### MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.

MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.

### Dimension. Eigenvalue and eigenvector

Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,

### Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

### 1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det

What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix

### PRACTICE PROBLEMS FOR THE FINAL

PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show

### ICS 6N Computational Linear Algebra Vector Space

ICS 6N Computational Linear Algebra Vector Space Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 24 Vector Space Definition: A vector space is a non empty set V of

### 2. Every linear system with the same number of equations as unknowns has a unique solution.

1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

### Worksheet for Lecture 15 (due October 23) Section 4.3 Linearly Independent Sets; Bases

Worksheet for Lecture 5 (due October 23) Name: Section 4.3 Linearly Independent Sets; Bases Definition An indexed set {v,..., v n } in a vector space V is linearly dependent if there is a linear relation

### MATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)

MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m

### 1. TRUE or FALSE. 2. Find the complete solution set to the system:

TRUE or FALSE (a A homogenous system with more variables than equations has a nonzero solution True (The number of pivots is going to be less than the number of columns and therefore there is a free variable

### (i) [7 points] Compute the determinant of the following matrix using cofactor expansion.

Question (i) 7 points] Compute the determinant of the following matrix using cofactor expansion 2 4 2 4 2 Solution: Expand down the second column, since it has the most zeros We get 2 4 determinant = +det

### MAT 242 CHAPTER 4: SUBSPACES OF R n

MAT 242 CHAPTER 4: SUBSPACES OF R n JOHN QUIGG 1. Subspaces Recall that R n is the set of n 1 matrices, also called vectors, and satisfies the following properties: x + y = y + x x + (y + z) = (x + y)

### Linear independence, span, basis, dimension - and their connection with linear systems

Linear independence span basis dimension - and their connection with linear systems Linear independence of a set of vectors: We say the set of vectors v v..v k is linearly independent provided c v c v..c

### Linear Independence x

Linear Independence A consistent system of linear equations with matrix equation Ax = b, where A is an m n matrix, has a solution set whose graph in R n is a linear object, that is, has one of only n +

### LINEAR ALGEBRA SUMMARY SHEET.

LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized

### March 27 Math 3260 sec. 56 Spring 2018

March 27 Math 3260 sec. 56 Spring 2018 Section 4.6: Rank Definition: The row space, denoted Row A, of an m n matrix A is the subspace of R n spanned by the rows of A. We now have three vector spaces associated

### Worksheet for Lecture 15 (due October 23) Section 4.3 Linearly Independent Sets; Bases

Worksheet for Lecture 5 (due October 23) Name: Section 4.3 Linearly Independent Sets; Bases Definition An indexed set {v,..., v n } in a vector space V is linearly dependent if there is a linear relation

### Linear Algebra: Sample Questions for Exam 2

Linear Algebra: Sample Questions for Exam 2 Instructions: This is not a comprehensive review: there are concepts you need to know that are not included. Be sure you study all the sections of the book and

### Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam

Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system

### Sept. 26, 2013 Math 3312 sec 003 Fall 2013

Sept. 26, 2013 Math 3312 sec 003 Fall 2013 Section 4.1: Vector Spaces and Subspaces Definition A vector space is a nonempty set V of objects called vectors together with two operations called vector addition

### Chapter 1. Vectors, Matrices, and Linear Spaces

1.6 Homogeneous Systems, Subspaces and Bases 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.6. Homogeneous Systems, Subspaces and Bases Note. In this section we explore the structure of the solution

### Third Midterm Exam Name: Practice Problems November 11, Find a basis for the subspace spanned by the following vectors.

Math 7 Treibergs Third Midterm Exam Name: Practice Problems November, Find a basis for the subspace spanned by the following vectors,,, We put the vectors in as columns Then row reduce and choose the pivot

### Elementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th.

Elementary Linear Algebra Review for Exam Exam is Monday, November 6th. The exam will cover sections:.4,..4, 5. 5., 7., the class notes on Markov Models. You must be able to do each of the following. Section.4

### Row Space and Column Space of a Matrix

Row Space and Column Space of a Matrix 1/18 Summary: To a m n matrix A = (a ij ), we can naturally associate subspaces of K n and of K m, called the row space of A and the column space of A, respectively.

### GENERAL VECTOR SPACES AND SUBSPACES [4.1]

GENERAL VECTOR SPACES AND SUBSPACES [4.1] General vector spaces So far we have seen special spaces of vectors of n dimensions denoted by R n. It is possible to define more general vector spaces A vector

### MATH 2360 REVIEW PROBLEMS

MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1

### 1 Last time: inverses

MATH Linear algebra (Fall 8) Lecture 8 Last time: inverses The following all mean the same thing for a function f : X Y : f is invertible f is one-to-one and onto 3 For each b Y there is exactly one a

### Summer Session Practice Final Exam

Math 2F Summer Session 25 Practice Final Exam Time Limit: Hours Name (Print): Teaching Assistant This exam contains pages (including this cover page) and 9 problems. Check to see if any pages are missing.

### Dr. Abdulla Eid. Section 4.2 Subspaces. Dr. Abdulla Eid. MATHS 211: Linear Algebra. College of Science

Section 4.2 Subspaces College of Science MATHS 211: Linear Algebra (University of Bahrain) Subspaces 1 / 42 Goal: 1 Define subspaces. 2 Subspace test. 3 Linear Combination of elements. 4 Subspace generated

### Vector space and subspace

Vector space and subspace Math 112, week 8 Goals: Vector space, subspace, span. Null space, column space. Linearly independent, bases. Suggested Textbook Readings: Sections 4.1, 4.2, 4.3 Week 8: Vector

### Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces. Section 1: Linear Independence

Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces Section 1: Linear Independence Recall that every row on the left-hand side of the coefficient matrix of a linear system A x = b which could

### (b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).

.(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)

### Math 240, 4.3 Linear Independence; Bases A. DeCelles. 1. definitions of linear independence, linear dependence, dependence relation, basis

Math 24 4.3 Linear Independence; Bases A. DeCelles Overview Main ideas:. definitions of linear independence linear dependence dependence relation basis 2. characterization of linearly dependent set using

### Assignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.

Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has

### Solutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015

Solutions to Final Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated /5/05 Answers This page contains answers only. See the following pages for detailed solutions. (. (a x. See

### Final Examination 201-NYC-05 December and b =

. (5 points) Given A [ 6 5 8 [ and b (a) Express the general solution of Ax b in parametric vector form. (b) Given that is a particular solution to Ax d, express the general solution to Ax d in parametric

### MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION

MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether

### IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

### Solving a system by back-substitution, checking consistency of a system (no rows of the form

MATH 520 LEARNING OBJECTIVES SPRING 2017 BROWN UNIVERSITY SAMUEL S. WATSON Week 1 (23 Jan through 27 Jan) Definition of a system of linear equations, definition of a solution of a linear system, elementary

### Review for Chapter 1. Selected Topics

Review for Chapter 1 Selected Topics Linear Equations We have four equivalent ways of writing linear systems: 1 As a system of equations: 2x 1 + 3x 2 = 7 x 1 x 2 = 5 2 As an augmented matrix: ( 2 3 ) 7

### MATH 1553, SPRING 2018 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9

MATH 155, SPRING 218 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9 Name Section 1 2 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 1 points. The maximum score

### 1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?

. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in

### Math 22 Fall 2018 Midterm 2

Math 22 Fall 218 Midterm 2 October 23, 218 NAME: SECTION (check one box): Section 1 (S. Allen 12:5) Section 2 (A. Babei 2:1) Instructions: 1. Write your name legibly on this page, and indicate your section

### MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)

### Worksheet for Lecture 23 (due December 4) Section 6.1 Inner product, length, and orthogonality

Worksheet for Lecture (due December 4) Name: Section 6 Inner product, length, and orthogonality u Definition Let u = u n product or dot product to be and v = v v n be vectors in R n We define their inner

### Chapter 2: Matrix Algebra

Chapter 2: Matrix Algebra (Last Updated: October 12, 2016) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). Write A = 1. Matrix operations [a 1 a n. Then entry

### Lecture 22: Section 4.7

Lecture 22: Section 47 Shuanglin Shao December 2, 213 Row Space, Column Space, and Null Space Definition For an m n, a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn, the vectors r 1 = [ a 11 a 12 a 1n

### Lecture 13: Row and column spaces

Spring 2018 UW-Madison Lecture 13: Row and column spaces 1 The column space of a matrix 1.1 Definition The column space of matrix A denoted as Col(A) is the space consisting of all linear combinations

### IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

### (c)

1. Find the reduced echelon form of the matrix 1 1 5 1 8 5. 1 1 1 (a) 3 1 3 0 1 3 1 (b) 0 0 1 (c) 3 0 0 1 0 (d) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (e) 1 0 5 0 0 1 3 0 0 0 0 Solution. 1 1 1 1 1 1 1 1

### Eigenvalues, Eigenvectors, and Diagonalization

Math 240 TA: Shuyi Weng Winter 207 February 23, 207 Eigenvalues, Eigenvectors, and Diagonalization The concepts of eigenvalues, eigenvectors, and diagonalization are best studied with examples. We will

### Section 4.5. Matrix Inverses

Section 4.5 Matrix Inverses The Definition of Inverse Recall: The multiplicative inverse (or reciprocal) of a nonzero number a is the number b such that ab = 1. We define the inverse of a matrix in almost

### Solutions to Math 51 First Exam April 21, 2011

Solutions to Math 5 First Exam April,. ( points) (a) Give the precise definition of a (linear) subspace V of R n. (4 points) A linear subspace V of R n is a subset V R n which satisfies V. If x, y V then

### Chapter 5. Eigenvalues and Eigenvectors

Chapter 5 Eigenvalues and Eigenvectors Section 5. Eigenvectors and Eigenvalues Motivation: Difference equations A Biology Question How to predict a population of rabbits with given dynamics:. half of the

### MATH10212 Linear Algebra B Homework 7

MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments

### 1. Let A = (a) 2 (b) 3 (c) 0 (d) 4 (e) 1

. Let A =. The rank of A is (a) (b) (c) (d) (e). Let P = {a +a t+a t } where {a,a,a } range over all real numbers, and let T : P P be a linear transformation dedifined by T (a + a t + a t )=a +9a t If

### Math 2331 Linear Algebra

4.3 Linearly Independent Sets; Bases Math 233 Linear Algebra 4.3 Linearly Independent Sets; Bases Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math233

### Math 4377/6308 Advanced Linear Algebra

2. Linear Transformations Math 4377/638 Advanced Linear Algebra 2. Linear Transformations, Null Spaces and Ranges Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/

### Math 415 Exam I. Name: Student ID: Calculators, books and notes are not allowed!

Math 415 Exam I Calculators, books and notes are not allowed! Name: Student ID: Score: Math 415 Exam I (20pts) 1. Let A be a square matrix satisfying A 2 = 2A. Find the determinant of A. Sol. From A 2

### University of Ottawa

University of Ottawa Department of Mathematics and Statistics MAT 1302A: Mathematical Methods II Instructor: Hadi Salmasian Final Exam April 2016 Surname First Name Seat # Instructions: (a) You have 3

### MATH 300, Second Exam REVIEW SOLUTIONS. NOTE: You may use a calculator for this exam- You only need something that will perform basic arithmetic.

MATH 300, Second Exam REVIEW SOLUTIONS NOTE: You may use a calculator for this exam- You only need something that will perform basic arithmetic. [ ] [ ] 2 2. Let u = and v =, Let S be the parallelegram

### Math 314H EXAM I. 1. (28 points) The row reduced echelon form of the augmented matrix for the system. is the matrix

Math 34H EXAM I Do all of the problems below. Point values for each of the problems are adjacent to the problem number. Calculators may be used to check your answer but not to arrive at your answer. That

### Row Space, Column Space, and Nullspace

Row Space, Column Space, and Nullspace MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Every matrix has associated with it three vector spaces: row space

### 6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if. (a) v 1,, v k span V and

6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if (a) v 1,, v k span V and (b) v 1,, v k are linearly independent. HMHsueh 1 Natural Basis

### Exam in TMA4110 Calculus 3, June 2013 Solution

Norwegian University of Science and Technology Department of Mathematical Sciences Page of 8 Exam in TMA4 Calculus 3, June 3 Solution Problem Let T : R 3 R 3 be a linear transformation such that T = 4,

### Math 308 Practice Test for Final Exam Winter 2015

Math 38 Practice Test for Final Exam Winter 25 No books are allowed during the exam. But you are allowed one sheet ( x 8) of handwritten notes (back and front). You may use a calculator. For TRUE/FALSE

### Test 3, Linear Algebra

Test 3, Linear Algebra Dr. Adam Graham-Squire, Fall 2017 Name: I pledge that I have neither given nor received any unauthorized assistance on this exam. (signature) DIRECTIONS 1. Don t panic. 2. Show all

### LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for

### Shorts

Math 45 - Midterm Thursday, October 3, 4 Circle your section: Philipp Hieronymi pm 3pm Armin Straub 9am am Name: NetID: UIN: Problem. [ point] Write down the number of your discussion section (for instance,

### General Vector Space (3A) Young Won Lim 11/19/12

General (3A) /9/2 Copyright (c) 22 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version